Modular supercuspidal lifts of weight 2

Abstract

Let $F/Q$ be a totally real number field and $N$ an ideal of its ring of integers of norm N. Let $p>\max \{k+1,6\}$ be a prime totally split in F such that $p\nmid N$. For every even $k\ge 2$, define the $[F:Q]$-dimensional parallel weight $\text{k}=(k,...,k)$. Let $f\in S_{\text{k}}\left({\Gamma }_0\right(N\left)\right)$ be any non CM Hilbert cuspidal Hecke eigenform. Assume that the residual representation ${\bar{\rho }}_{f,P}$ has large image for some prime $P$ over p in the field of definition of f. Under these conditions, we prove that there exists a lift of ${\bar{\rho }}_{f,P}$ associated to a Hilbert modular cuspform $g\in S_{\text{2}}(N{p}^2,ϵ)$ which is supercuspidal at each prime of F over p. We also give a proof of the corresponding statement for classical Hecke cuspforms. Such statement was already proved by Khare [23] with classical techniques. Finally, using our main result we give a corrigenda for [12], correctly inserting the micro good dihedral prime in the level.